Fracture mechanics studies the initiation and growth of cracks under loading. It connects a defect's geometry with the local driving force for crack extension and the material's resistance. A component's average stress alone cannot describe the concentration of deformation near a crack tip.[ref]
In linear elastic fracture mechanics, a mode-I stress intensity factor often has the form , where is a specified crack dimension and accounts for geometry. Doubling increases by at fixed nominal stress. Comparing the appropriate intensity with measured fracture toughness is meaningful only when the loading, geometry and small-scale yielding assumptions of the model apply.
Linear elastic fracture mechanics predicts the loading of a crack using linear elasticity outside a small region near its tip. The near-tip stress field has a characteristic dependence; its amplitude is the stress intensity factor. For an opening crack under nominal tensile stress ,where is a specified crack-length parameter and a dimensionless geometry factor. A central crack in an effectively infinite plate has total length and . Crack advance is assessed by comparing with the appropriate measured fracture toughness.[ref]
For and in that ideal geometry, . Increasing the crack's half-length fourfold doubles without changing the applied stress. This explains why a structure can fail at a stress below the strength of an uncracked specimen.
A toughness measurement loads a precracked specimen of known geometry and records the force associated with crack extension. Thickness, remaining ligament and crack length must be large relative to the crack-tip plastic region before the plane-strain value is transferable in this model. Large-scale plastic deformation requires a different description; the elastic singularity is a useful outer approximation, not an assertion of physically infinite stress inside the material.[ref]
Introduction to Fracture Mechanics by David Roylance. printed pp.9–11, specimen geometry, plastic zone and plane strain
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Fracture mechanics is a branch of mechanics that studies the behavior of materials containing cracks or flaws. It aims to understand how and why materials fail when they are subjected to stress, and it helps in predicting the conditions under which a crack will grow, leading to the failure of a structure or component. The primary focus of fracture mechanics is on the propagation of cracks and the factors that influence that propagation.